Don’t worry, I won’t kill you. I’ll just have my hitman do it.

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It’s called a chasuble. In the picture you have shown, he is wearing a Roman chasuble. Some traditional priests still wear Roman chasubles.
During a Tridentine Mass, the deacon (if it’s a High Mass) raises the priest’s chasuble. The altar server does it if it’s a Low Mass. The chasuble represents the Yoke of Christ and charity. By lifting it, you are symbolizing that the Yoke of Christ and charity is open to everyone, not just clergy. I don’t believe the raising of the chasuble is required in a Low Mass, through it is often done. Below is a link if you want to know more about it.
arsorandi.blogspot.com/2010/10/symbolic-purpose-of-lifting-rim-of.html
God bless.
P.S. I love the Tridentine Mass!